Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprRange, Lambda, Variable, VertExprArray, e, l
from proveit.logic import And, Equals, InSet, Union
from proveit.numbers import Add, Interval, ModAbs, greater, one
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import ModAdd, QPE, _U, _b_floor, _full_domain, _ket_u, _neg_domain, _pos_domain, _s, _s_wire, _t, _two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(_t, one)
sub_expr3 = Add(_t, _s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr6 = greater(ModAbs(l, _two_pow_t), e)
sub_expr7 = Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)], [ExprRange(sub_expr1, sub_expr5, one, _t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(ModAdd(_b_floor, l), _t), part = sub_expr1), targets = Interval(one, _t)), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]))
expr = Equals(Lambda(l, Conditional(sub_expr7, And(InSet(l, Union(_neg_domain, _pos_domain)), sub_expr6))), Lambda(l, Conditional(sub_expr7, And(InSet(l, _full_domain), sub_expr6)))).with_wrapping_at(2)

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} \left[l \mapsto \left\{\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array} \textrm{ if } l \in \left(\{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\} \cup \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right) ,  \left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right..\right] =  \\ \left[l \mapsto \left\{\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} ,  \left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right..\right] \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Lambda	parameter: 127 body: 5
4	Lambda	parameter: 127 body: 7
5	Conditional	value: 9 condition: 8
6	ExprTuple	127
7	Conditional	value: 9 condition: 10
8	Operation	operator: 14 operands: 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16, 22
12	Literal
13	ExprTuple	17, 18, 19, 20
14	Literal
15	ExprTuple	21, 22
16	Operation	operator: 32 operands: 23
17	ExprTuple	24, 25
18	ExprTuple	26, 27
19	ExprTuple	28, 29
20	ExprTuple	30, 31
21	Operation	operator: 32 operands: 33
22	Operation	operator: 34 operands: 35
23	ExprTuple	127, 36
24	ExprRange	lambda_map: 37 start_index: 136 end_index: 132
25	ExprRange	lambda_map: 38 start_index: 136 end_index: 119
26	ExprRange	lambda_map: 39 start_index: 136 end_index: 132
27	ExprRange	lambda_map: 39 start_index: 105 end_index: 106
28	ExprRange	lambda_map: 40 start_index: 136 end_index: 132
29	ExprRange	lambda_map: 41 start_index: 136 end_index: 119
30	ExprRange	lambda_map: 42 start_index: 136 end_index: 132
31	ExprRange	lambda_map: 43 start_index: 136 end_index: 119
32	Literal
33	ExprTuple	127, 44
34	Literal
35	ExprTuple	121, 45
36	Operation	operator: 46 operands: 47
37	Lambda	parameter: 104 body: 48
38	Lambda	parameter: 104 body: 49
39	Lambda	parameter: 104 body: 50
40	Lambda	parameter: 104 body: 51
41	Lambda	parameter: 104 body: 52
42	Lambda	parameter: 104 body: 53
43	Lambda	parameter: 104 body: 55
44	Operation	operator: 96 operands: 56
45	Operation	operator: 57 operands: 58
46	Literal
47	ExprTuple	59, 60
48	Operation	operator: 88 operands: 61
49	Operation	operator: 68 operands: 62
50	Operation	operator: 68 operands: 63
51	Operation	operator: 64 operands: 65
52	Operation	operator: 89 operands: 66
53	Operation	operator: 68 operands: 67
54	ExprTuple	104
55	Operation	operator: 68 operands: 69
56	ExprTuple	84, 120
57	Literal
58	ExprTuple	127, 70
59	Operation	operator: 96 operands: 71
60	Operation	operator: 96 operands: 72
61	NamedExprs	state: 73
62	NamedExprs	element: 74 targets: 82
63	NamedExprs	element: 75 targets: 76
64	Literal
65	NamedExprs	basis: 77
66	NamedExprs	operation: 78
67	NamedExprs	element: 79 targets: 80
68	Literal
69	NamedExprs	element: 81 targets: 82
70	Operation	operator: 124 operands: 83
71	ExprTuple	84, 85
72	ExprTuple	108, 120
73	Operation	operator: 86 operand: 100
74	Operation	operator: 88 operands: 95
75	Operation	operator: 89 operands: 90
76	Operation	operator: 96 operands: 91
77	Literal
78	Literal
79	Operation	operator: 94 operands: 92
80	Operation	operator: 96 operands: 93
81	Operation	operator: 94 operands: 95
82	Operation	operator: 96 operands: 97
83	ExprTuple	128, 132
84	Operation	operator: 130 operands: 98
85	Operation	operator: 134 operand: 108
86	Literal
87	ExprTuple	100
88	Literal
89	Literal
90	NamedExprs	operation: 101 part: 104
91	ExprTuple	136, 106
92	NamedExprs	state: 102 part: 104
93	ExprTuple	136, 132
94	Literal
95	NamedExprs	state: 103 part: 104
96	Literal
97	ExprTuple	105, 106
98	ExprTuple	107, 136
99	ExprTuple	108
100	Literal
101	Operation	operator: 109 operands: 110
102	Operation	operator: 111 operands: 112
103	Literal
104	Variable
105	Operation	operator: 130 operands: 113
106	Operation	operator: 130 operands: 114
107	Operation	operator: 134 operand: 120
108	Operation	operator: 130 operands: 116
109	Literal
110	ExprTuple	117, 132
111	Literal
112	ExprTuple	118, 132
113	ExprTuple	132, 136
114	ExprTuple	132, 119
115	ExprTuple	120
116	ExprTuple	121, 136
117	Literal
118	Operation	operator: 122 operands: 123
119	Literal
120	Operation	operator: 124 operands: 125
121	Variable
122	Literal
123	ExprTuple	126, 127
124	Literal
125	ExprTuple	128, 129
126	Literal
127	Variable
128	Literal
129	Operation	operator: 130 operands: 131
130	Literal
131	ExprTuple	132, 133
132	Literal
133	Operation	operator: 134 operand: 136
134	Literal
135	ExprTuple	136
136	Literal

Expression of type Equals¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶